# Enneahedra with 11 vertices: 444444444 and 333345555

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay

General Notes on Polyhedron Enumeration

## 444444444

Although this class has only one representative, it's interesting because it's the simplest polyhedron with an odd number of faces, all quadrilaterals. It consists of a central band of three quadrilaterals, joined corner to corner, and two pyramidal caps. If the central faces are rhombi, the solid has axial threefold symmetry and mirror planes and equatorial mirror symmetry. Despite its simplicity, or perhaps because of it, the planar nets don't show the structure very well. In the nets above, the equatorial faces are colored darker than the polar caps. Because of the high degree of symmetry, there are only two distinct representations, one with an equatorial face as base, the other with a pyramid face.

It's impossible to construct this solid with all rhombic faces. The two polar caps consist of three kite-shaped polygons. The lower diagram shows the dimensions. A polar view of the solid is shown at lower left. The vertical axis of each rhombic face is 2h and the horizontal axis is 2. Since ac = ad, and ab= ac/3, then db = 4/3 ad, and the height of the apex is 4/3 h. An equatorial view is shown at lower right.

The short diagonal of the kite has length 1 and crosses the long axis 3/4 of the way from the acute end to the obtuse end. The long axis of each kite face in the polar view is 2sqrt(3)/3. The vertical distance from the plane of the diagram to the apex is 4h/3. The total length is thus sqrt(4/3 + 16h2/9) = 2sqrt(3 + 4h2)/3.

## 333345555

This class has 35 representatives. The unique quadrilateral face suggests using that face as a base might allow the best view. Unfortunately, interior faces are often tiny in this view. Each polyhedron is shown in all four pentagonal base views as well as the quadrilateral face view.

004 (below) is the only form where the quadrilateral is completely surrounded by pentagons.

001, 010 and 012 (below) have a quadrilateral face with pentagons on three sides. All four triangular faces are in contact.

030 (above) and 032, 031, 013, and 014 have a quadrilateral face with pentagons on three sides, and three triangle faces in side-by-side contact.